We develop a new approach to cluster algebras, based on the notion of an upper cluster algebra defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in [7], we show that under an assumption of ``acyclicity,'' a cluster algebra coincides with its upper counterpart and is finitely generated; in this case, we also describe its defining ideal and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to an upper cluster algebra explicitly defined in terms of relevant combinatorial data.

We define a renormalized characteristic class for Einstein asymptotically complex hyperbolic (ACHE) manifolds of dimension 4: for any such manifold, the polynomial in the curvature associated to the characteristic class χ−3τ is shown to converge. This extends a work of Burns and Epstein in the Kähler-Einstein case

We also define a new global invariant for any compact 3-dimensional strictly pseudoconvex Cauchy-Riemann (CR) manifold by a renormalization procedure of the η-invariant of a sequence of metrics that approximate the CR structure.

Finally, we get a formula relating the renormalized characteristic class to the topological number χ−3τ and the invariant of the CR structure arising at infinity.

We give two proofs of a conjecture of Hori and Vafa which expresses the J-function—a generating function for 1-point descendant Gromov-Witten invariants—of a Grassmannian in terms of the J-function of a product of projective spaces. Similar relations are obtained for 2-point descendants and 3-point primary Gromov-Witten invariants. As an application, we prove Givental's R-conjecture, and hence the Virasoro conjecture, for Grassmannians.

We give a decomposition of the Chow motive of an isotropic projective homogeneous variety of a semisimple algebraic group in terms of twisted motives of simpler projective homogeneous varieties. As an application, we prove a generalization of Rost's nilpotence theorem.

La correspondance de Jacquet-Langlands établit une bijection Hecke-équivariante entre les espaces de formes modulaires quaternioniques et certains espaces de formes modulaires usuelles. Dans cet article, nous montrons qu'elle se prolonge en un isomorphisme rigide analytique entre des courbes de Hecke définies de part et d'autre, de sorte qu'elle s'étend aux formes p-adiques surconvergentes de pente finie, ainsi qu'aux familles p-adiques.

In this paper, we extend the Jacquet-Langlands correspondence between Hecke-modules of usual modular forms and quaternionic modular forms to overconvergent p-adic forms of finite slope. We show that this correspondence respects p-adic families and is induced by an isomorphism between some associated eigencurves.